Question #278882

Solve the following triangles:


13. A side and a diagonal of a parallelogram are 12 inches and 19 inches respectively. The angle between the diagonals, opposite the given side, is 124°. Find the length of the other diagonal and the length of the other side.

14. A lighthouse is 10 miles northeast of a dock. A ship leaves the dock at noon, and sails east at a speed of 12 miles an hour. At what time will it be 8 miles from the lighthouse?

15.A vertical pole 35 feet high, standing on sloping ground, is braced by a wire which extends from the top of the pole to a point on the ground 25 feet from the foot of the pole. If the pole. subtends an angle of 30° at the point where the wire reaches the ground, how long is the wire?

16.A tower 125 feet high stands on the side of a hill. At a point 240 feet from the foot of the tower, measured straight down the hill, the tower subtends an angle of 25°. What angle does the side of the hill make with the horizontal?


1
Expert's answer
2021-12-13T17:31:33-0500

(13)

Let b== Side adjacent to angle of 124°124°

Using cosine rule


122=9.52+b2=2(b)(9.5)cos12412^2=9.5^2+b^2=2(b)(9.5) cos 124

b2+10.62b53.75=0b^2+10.62b-53.75=0



b=10.62+10.622+4(53.75)2b=\frac{-10.62{^+_-}\sqrt{10.62^2+4(53.75)}}{2}


b=3.74b=3.74 or 14.36-14.36

Length of the diagonal =3.74×2=3.74×2

=7.48inches=7.48inches


(14)


Let b == distance sailed.

Using cosine Rule


82=b2+1022×10×bcos458^2=b^2+10^2-2×10×b\>cos\>45

b214.14b+36=0b^2-14.14b+36=0


b=14.14+14.1424(36)2b=\frac{14.14{^+_-}\sqrt{14.14^2-4(36)}}{2}

=3.329or10.813=3.329\>or\>10.813



Time taken =3.32912×60=1639=\frac{3.329}{12}×60=16'39''


Or 10.81312×60=544\frac{10.813}{12}×60=54'4''


Time can be;;

12:1639p.m12:16'39''\>p.m

Or12:544p.mOr\>12:54'4''\>p.m


(15)


Let b== length of the wire

Using cosine rule


352=b2+2522×b×25cos3035^2=b^2+25^2-2×b×25\>cos\>30


b243.30b600=0b^2-43.30b-600=0


b=43.30+43.324(600)2b=\frac{43.30{^+_-}\sqrt{43.3^2-4(-600)}}{2}


=54.34or11.04=54.34\>or\>-11.04

Length of the wire =54.34feet=54.34feet



(16)


From a triangle 240 feet opposite to °\empty° and 125 feet opposite to 25°25°


\therefore 240Sin=125Sin25\frac{240}{Sin{\empty}}=\frac{125}{Sin25}


Sin=240Sin25125Sin{\empty}=\frac{240\>Sin25}{125}


=54.24°{\empty}=54.24°


Let θ\theta== angle to horizontal

θ=90(54.24+25)\theta=90-(54.24+25)

=10.76°=10.76°


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